Question: $ B = \left[\begin{array}{rr}2 & 5 \\ 4 & -1\end{array}\right]$ $ C = \left[\begin{array}{rrr}5 & 3 & -1 \\ 4 & 5 & 2\end{array}\right]$ What is $ B C$ ?
Solution: Because $ B$ has dimensions $(2\times2)$ and $ C$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ B C = \left[\begin{array}{rr}{2} & {5} \\ {4} & {-1}\end{array}\right] \left[\begin{array}{rrr}{5} & \color{#DF0030}{3} & \color{#9D38BD}{-1} \\ {4} & \color{#DF0030}{5} & \color{#9D38BD}{2}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ B$ , with the corresponding elements in column $j$ of the second matrix, $ C$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ B$ with the first element in ${\text{column }1}$ of $ C$ , then multiply the second element in ${\text{row }1}$ of $ B$ with the second element in ${\text{column }1}$ of $ C$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{2}\cdot{5}+{5}\cdot{4} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ B$ with the corresponding elements in ${\text{column }1}$ of $ C$ and add the products together. $ \left[\begin{array}{rrr}{2}\cdot{5}+{5}\cdot{4} & ? & ? \\ {4}\cdot{5}+{-1}\cdot{4} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ B$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ C$ and add the products together. $ \left[\begin{array}{rrr}{2}\cdot{5}+{5}\cdot{4} & {2}\cdot\color{#DF0030}{3}+{5}\cdot\color{#DF0030}{5} & ? \\ {4}\cdot{5}+{-1}\cdot{4} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{2}\cdot{5}+{5}\cdot{4} & {2}\cdot\color{#DF0030}{3}+{5}\cdot\color{#DF0030}{5} & {2}\cdot\color{#9D38BD}{-1}+{5}\cdot\color{#9D38BD}{2} \\ {4}\cdot{5}+{-1}\cdot{4} & {4}\cdot\color{#DF0030}{3}+{-1}\cdot\color{#DF0030}{5} & {4}\cdot\color{#9D38BD}{-1}+{-1}\cdot\color{#9D38BD}{2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}30 & 31 & 8 \\ 16 & 7 & -6\end{array}\right] $